The Lie algebra of a Lie group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 12 August 2013

Lie algebras and the exponential map

A Lie group is a group that is also a manifold, i.e. a topological group that is locally isomorphic to ${ℝ}^n$.

$$1 U G φ:U→∼V ℝn 0 V U is an openneighbourhood of 1 in G V is an openneighbourhood of 0 in ℝn$$ If $G$ is connected then $G$ is generated by the elements of $U$.

The exponential map is a smooth homomorphism $$\begin{array}{ccc}𝔤& \to & G\\ 0& \mapsto & 1\end{array}$$ which is a homeomorphism on a neighborhood of $0$.

The Lie algebra $𝔤$ contains the structure of $G$ in a neighbourhood of the identity.

A one parameter subgroup of $G$ is a smooth group homomorphism $\gamma :ℝ\to G$.

$$0 t 1 γ(t) ⟼γ$$

Examples of one-parameter subgroups:

1. Define $$\begin{array}{rcl}\gamma :ℝ& \to & {\mathrm{GL}}_n\left(ℝ\right)\\ t& \mapsto & 1+t{E}_{ij}=x_{ij}\left(t\right),\text{for}i\ne j.\end{array}$$ Note that $$x_{ij}\left(t\right)x_{ij}\left(s\right)=x_{ij}(s+t)$$ since $$\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& s\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1& t+s\\ 0& 1\end{array}\right).$$
2. Define $$\begin{array}{rrcl}\gamma :& ℝ& \to & {\mathrm{GL}}_n\left(ℝ\right)\\ & t& \mapsto & \left(\begin{array}{ccccccc}1& & & & & & \\ & \ddots & & & & & \\ & & 1& & & & \\ & & & e^t& & & \\ & & & & 1& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right)=h_i\left(e^t\right).\end{array}$$ Note that $$h_i\left(e^t\right)h_i\left(e^s\right)=h_i\left(e^{t+s}\right).$$

Let $G$ be a Lie group. The ring of functions on $G$ is $$C^{\infty }\left(G\right)=\{f:G\to ℝ|f\text{is smooth at}g\text{for all}g\in G\}$$ where $$f\text{is smooth at}g\text{if}{ \frac{d^{k}f}{d{x}^k}| }_{x=g}\text{exists for all}k\in {\mathbb{Z}}_{>0}.$$

Let $g\in G$.

• A tangent vector at $g$ is a linear map $\eta :C^{\infty }\left(G\right)\to ℝ$ such that $$\eta \left(f_1{f}_2\right)=f_1\left(g\right)\eta \left(f_2\right)+\eta \left(f_1\right)f_2\left(g\right),\text{for}{f}_1,f_2\in C^{\infty }\left(G\right).$$
• A vector field on $G$ is a linear map $\partial :C^{\infty }\left(G\right)\to C^{\infty }\left(G\right)$ such that $$\partial \left(f_1{f}_2\right)=f_1\partial \left(f_2\right)+\partial \left(f_1\right)f_2,\text{for}{f}_1,f_2\in C^{\infty }\left(G\right).$$
• A left invariant vector field on $G$ is a vector field $\partial :C^{\infty }\left(G\right)\to C^{\infty }\left(G\right)$ such that $$L_g\partial =\partial L_g,\text{for all}g\in G,$$ where $L_g:C^{\infty }\left(G\right)\to C^{\infty }\left(G\right)$ is given by $$\left(L_{g}f\right)\left(x\right)=f\left(g^{-1}x\right),\text{for}f\in C^{\infty }\left(G\right),g,x\in G.$$
• A one-parameter subgroup of $G$ is a smooth group homomorphism $\gamma :ℝ\to G$.
• The Lie algebra $𝔤=\mathrm{Lie}\left(G\right)$ of $G$ is the vector space of left invariant vector fields on $G$ with bracket $$[{\partial }_1,{\partial }_2]={\partial }_1{\partial }_2-{\partial }_2{\partial }_1,\text{for}{\partial }_1,{\partial }_2\in 𝔤.$$

If $\gamma$ is a one-parameter subgroup of $G$ define $$\frac{df\left(\gamma \right(t\left)\right)}{dt}=\underset{h\to 0}{\lim }\frac{f\left(\gamma \right(t+h\left)\right)-f\left(\gamma \right(t\left)\right)}{h}.$$ The following proposition says that we can identify three vector spaces

1. {left invariant vector fields on $G$},
2. {one parameter subgroups of $G$},
3. {tangent vectors at $1\in G$}.

The maps $$\begin{array}{rcl}\text{{left invariant vector fields}}& \to & \text{{tangent vectors at 1}}\\ \xi & \mapsto & {\xi }_1\end{array}$$ and $$\begin{array}{rcl}\text{{one parameter subgroups}}& \to & \text{{tangent vectors at 1}}\\ \gamma & \mapsto & {\gamma }_1\end{array}$$ where $${\xi }_{1}f=\left(\xi f\right)\left(1\right),\text{and}{\gamma }_1=\left(\frac{d}{dt}f\left(\gamma \left(t\right)\right)\right){|}_{t=0},$$ are vector space isomorphisms.

The exponential map is $$\begin{array}{rcl}𝔤& \to & G\\ tX& \mapsto & e^{tX}& \text{where}{e}^{tX}=\gamma \left(t\right),\end{array}$$ where $\gamma$ is the one-parameter subgroup corresponding to the tangent vector $X$.

Examples of exponential maps:

1. The Lie algebra ${\mathrm{𝔤𝔩}}_n$ is $${\mathrm{𝔤𝔩}}_n=\{x\in M_n(ℂ\left)\right\}\text{with bracket}[x_1,x_2]=x_1{x}_2-x_2{x}_1.$$ The vector space $${\mathrm{𝔤𝔩}}_n\text{has basis}\left\{E_{ij}\right|1\le i,j\le n\},$$ where $E_{ij}$ is the matrix with $1$ in the $(i,j)$-entry and 0 elsewhere. The exponential map is $$\begin{array}{ccc}{\mathrm{𝔤𝔩}}_n& \to & {\mathrm{GL}}_n\\ tX& \mapsto & e^{tX},\end{array}$$ where $$e^A=1+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots$$ for a matrix $A$. Then $$e^{t{E}_{ij}}=1+t{E}_{ij}=\left(\begin{array}{cccc}1& & & \\ & \ddots & t& \\ & & \ddots & \\ & & & 1\end{array}\right)=x_{ij}\left(t\right),\text{for}i\ne j,$$ where $t$ is in the $(i,j)$ matrix entry, and $$e^{t{E}_{ii}}=\left(\begin{array}{ccccccc}1& & & & & & \\ & \ddots & & & & & \\ & & 1& & & & \\ & & & e^t& & & \\ & & & & 1& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right)=h_i\left(e^t\right).$$
2. If $n=1$ the exponential map $$\begin{array}{ccc}ℂ& \to & {ℂ}^{\times }\\ tx& \mapsto & e^{tx}\end{array}$$ is a homeomorphism from a neighbourhood of $0$ to a neighbourhood of $1$. In fact, if $e\left(t\right)=a_0+a_{1}t+a_2{t}^2+\cdots$ and $$e(s+t)=e\left(s\right)e\left(t\right),$$ then $$\begin{array}{rcl}e(s+t)& =& a_0+a_1(s+t)+a_2(s+t{)}^2+a_3(s+t{)}^3+\cdots \\ & =& \begin{array}{c}{a}_0+\\ a_{1}s+a_{1}t+\\ a_2{s}^2+2a_{2}st+a_2{t}^2\\ a_3{s}^3+3a_3{s}^{2}t+3a_{3}s{t}^2+a_3{t}^3+\\ ⋮\end{array}\end{array}$$ and $$\begin{array}{rcl}e\left(s\right)e\left(t\right)& =& (a_0+a_{1}s+a_2{s}^2+a_3{s}^3+\cdots )(a_0+a_{1}t+a_2{t}^2+a_3{t}^3+\cdots )\\ & =& \begin{array}{c}{a}_0^2+\\ a_0{a}_{1}s+a_0{a}_{1}t+\\ a_0{a}_2{s}^2+a_1^{2}st+a_0{a}_2{t}^2\\ a_0{a}_3{s}^3+a_2{a}_1{s}^{2}t+a_1{a}_{2}s{t}^2+a_0{a}_3{t}^3+\\ ⋮\end{array}\end{array}$$ Hence $e(s+t)=e\left(s\right)e\left(t\right)$ only if $$a_0{a}_1=a_1,2a_2=a_1^2,3a_3=a_1{a}_2,4a_3=a_1{a}_3,\dots$$ so that $$a_0=1,a_2=\frac{a_1^2}{2},a_3=\frac{a_1^3}{3!},a_4=\frac{a_1^4}{4!},\dots$$ and $$e\left(t\right)=1+a_{1}t+\frac{a_1^2}{2}{t}^2+\frac{a_1^3}{3!}{t}^3+\cdots =e^{a_{1}t}.$$ So $$\begin{array}{ccc}ℂ& \to & {ℂ}^{\times }\\ z& \mapsto & e^z\end{array}$$ is the "unique" smooth homomorphism $ℂ\to {ℂ}^{\times }$.

The functor from Lie groups to Lie algebras

Let $\phi :G\to H$ be a Lie group homomorphism and let $𝔤=\mathrm{Lie}\left(G\right)$ and $𝔥=\mathrm{Lie}\left(H\right)$. The homomorphism $\phi$ corresponds to a morphisms between the rings of functions on $G$ and $H$, $$\begin{array}{rcl}{C}^{\infty }\left(H\right)& \stackrel{{\phi }^{*}}{\to }& C^{\infty }\left(G\right)\\ f& \mapsto & f\circ \phi \end{array}$$ and the differential of $\phi$ is the Lie group homomorphism $𝔤\stackrel{d\phi }{\to }𝔥$ given by $$\begin{array}{rcl}d\phi \left({\xi }_1\right)={\xi }_1\circ {\phi }^{*}& \text{if}& {\xi }_1\text{is a tangent vector at the identity,}\\ d\phi \left(\xi \right)=\xi \circ {\phi }^{*}& \text{if}& \xi \text{is a left invariant vector field,}\\ d\phi \left(\gamma \right)=\phi \circ \gamma & \text{if}& \gamma \text{is a one parameter subgroup.}\end{array}$$

(Note: It should be checked that

1. the map $d\phi$ is well defined,
2. the three definitions of $d\phi$ are the same,
3. and that $d\phi$ is a Lie algebra homomorphism.

These checks are not immediate, but are quite straightforward checks from the defiinitions.) The map $$\begin{array}{rcl}\text{the category of Lie groups}& \to & \text{the category of Lie algebras}\\ G& \mapsto & \mathrm{Lie}\left(G\right)\\ \phi & \mapsto & d\phi \end{array}$$ is a functor. This functor is not one to one; for example, the Lie groups $O_n\left(ℝ\right)$ and ${SO}_n\left(ℝ\right)$ have the same Lie algebra. On the other hand, the Lie algebra does contain the complete structure of the Lie group in a neighbourhood of the identity. The exponential map is $$\begin{array}{rcl}𝔤& \to & G\\ tX& \mapsto & e^{tX},\end{array}\text{where}{e}^{t}X=\gamma \left(t\right)$$ is the one parameter subgroup corresponding to $X\in 𝔤$. This map is a homeomorphism from a neighbourhood of 0 in $𝔤$ to a neighbourhood of 1 in $G$.

(Lie's theorem) The functor $$\begin{array}{rcll}\text{Lie:}& \text{{connected simply connected Lie groups}}& \to & \text{{Lie algebras}}\\ & G& \mapsto & 𝔤=\mathrm{Lie}\left(G\right)=T_1\left(G\right)\end{array}$$ is an equivalence of categories.

If $𝔤$ is a Lie subalgebra of $𝔤{𝔩}_n$ then the matrices $$\left\{e^{tX}\right|t\in ℝ,X\in 𝔤{𝔩}_n\},\text{where}{e}^{t}X=\sum _{k\ge 0}\frac{t^k{X}^k}{k!},$$ form a group with Lie algebra $𝔤$.

The following formulas relating expansions in the Lie group to expansions in the Lie algebra can be worked out by directly expanding and multipliying the power series for the exponentials or by using techniques such as those found in [BouLie, Ch. ????, Sect. 6]. $$e^{tX}{e}^{tY}=e^{t(X+Y)+(t^2/2)[X,Y]+\dots },$$ $$e^{tX}{e}^{tY}{e}^{-tX}=e^{tY+t^2[X,Y]+\dots },$$ $$e^{tX}{e}^{tY}{e}^{-tX}{e}^{-tY}=e^{t^2[X,Y]+\dots },$$

WHAT IS THE NICE one-line RELATION BETWEEN THE ASSOCIATIVE LAW IN THE GROUP AND THE JACOBI IDENTITY IN THE LIE ALGEBRA???

Relating representations of Lie groups and Lie algebras

If $V:G\to \mathrm{GL}\left(V\right)$ is a homomorphism of Lie groups then the differential of $V$ is a Lie algebra homomorphism $$dV:𝔤\to \mathrm{End}\left(V\right)$$ which satisfies $$dV\left(\right[x,y\left]\right)=\left[dV\right(x),dV(y\left)\right]=dV\left(x\right)dV\left(y\right)-dV\left(y\right)dV\left(x\right),$$ for $x,y\in 𝔤$. A representation of a Lie algebra $𝔤$, or a $𝔤$-module, is an action of $𝔤$ on a vector space $V$ by linear transformations, (a linear map $\phi :𝔤\to \mathrm{End}\left(V\right)$) such that $$V\left(\right[x,y\left]\right)=\left[V\right(x),V(y\left)\right]=V\left(x\right)V\left(y\right)-V\left(y\right)V\left(x\right),\text{for all}x,y\in 𝔤,$$ where $V\left(x\right)$ is the linear transformation on $V$determined by the action of $x\in 𝔤$. The trivial representation of $𝔤$ is the map $$\begin{array}{rcll}1:& 𝔤& \to & ℂ\\ & x& \mapsto & 0\end{array}$$ If $V$ is a $𝔤$-module, the dual $𝔤$-module is the $𝔤$-action on $V*=\mathrm{Hom}(V,ℂ)$ given by $$\left(x\phi \right)\left(v\right)=\phi \left(-xv\right),\text{for}x\in 𝔤,\phi \in V*,v\in V.$$ If $V$ and $W$ are $𝔤$-modules then the tensor product of $V$ and $W$ is the $𝔤$-action on $V\otimes W$ given by $$x(v\otimes w)=xv\otimes w+v\otimes xw,x\in 𝔤,v\in V,w\in W.$$

The definitions of the trivial, dual and tensor product $𝔤$-modules are accounted for by the following formulas: DOES THE SPACING OF THESE FORMULAS NEED TWEAKING?? $$\left(\frac{d}{dt}1\right){|}_{t=0}=\left(\frac{d}{dt}{e}^{t\cdot 0}\right){|}_{t=0}=0,$$ $$\left(\frac{d}{dt}{\left(e^{tX}\right)}^{-1}\right){|}_{t=0}=\left(\frac{d}{dt}(e^{-tX}-1)\right){|}_{t=0}=-X,$$ $$\left(\frac{d}{dt}(e^{tX}\otimes e^{tX})\right){|}_{t=0}=(\frac{d}{dt}\left(\right(1+tX+\frac{t^2{X}^2}{2!}+\dots )\otimes (1+tX+\frac{t^2{X}^2}{2!}+\dots )){|}_{t=0}=\left(\frac{d}{dt}(1\otimes 1+t(X\otimes 1+1\otimes X)+\dots )\right){|}_{t=0}=X\otimes 1+1\otimes X.$$

Notes and References

These notes are adapted from various lectures of Arun Ram on Representation theory, from 2008 and from Work2004/Book2003/chap41.17.03.pdf.

References

References?

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